Spectral problem for the Schrodinger equation with singular potential polynomial of even degree

For the stationary Schrodinger equation with singular potential U(x)=x^2r, where r∈Z₊, that describes situations associated with phase transition in quantum systems, discovery of symmetries can be used to reduce problems to algebraic problems for finding spectra. We construct analytic procedures for the convergent theory of perturbations to find eigenvalues for potentials of the indicated type. We discuss the fine structure of Bohr—Sommerfeld quantum conditions. Moscow State Institute of Radio Engineering, Electronics, and Automation. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 55–70, May, 1996.